Apparatus for altering the apparent perspective of images



March 15, 1966 H. s. HEMSTREET APPARATUS FOR ALTERING THE APPARENTPERSPECTIVE 0F IMAGES Original Filed Nov. 25, 1955 ll Sheets-Sheet 1 U ER Y R O E T m w W w Z o 2 M .N. m G s P w m 0 o M u H h L w 3 m 8 w t m2 L m 2 Q I M I n O m March 1966 H. s. HEMSTREET APPARATUS FOR ALTBRINGTHE APPARENT PERSPECTIVE OF IMAGES 11 Sheets-Sheet 2 Original Filed Nov.25, 1955 SERVO HORIZ SWEEP GEN FIG.9

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INVENTOR ATTORNEY March 15, 1966 H. s. HEMSTREET APPARATUS FOR ALTERINGTHE APPARENT PERSPECTIVE OF IMAGES 11 Sheets-Sheet 4 Original Filed Nov.25, 1955 HAROLD S. HEM STREET INVENTOR Wfl ATTORNEY March 15, 1966 H. s.HEMSTREET 3,240,120

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ATTORNEY H. S. HEMSTREET March 15, 1966 APPARATUS FOR ALTERING THEAPPARENT PERSPECTIVE OF IMAGES l1 Sheets-Sheet 11 Original Filed Nov.25, 1955 MIQV i u fw w I Non- D non- 207m honm mwmninTm 3 0-071 HAROLDS. HEMSTREET INVENTOR ATTORNEY United States Patent 3,240,120 APPARATUSFOR ALTERING THE APPARENT PERSPECTIVE 0F IMAGES Harold S. Hemstreet,Wilton, Conn., assignor to General Precision, Inc., a corporation ofDelaware Continuation of application Ser. No. 548,841, Nov. 25, 1955.This application Oct. 30, 1961, Ser. No. 155,227 25 Claims. (CI. 8857)This invention relates to apparatus for altering the apparentperspective of planar images, and is a continuation of my copendingapplication Serial Number 548,841, filed November 25, 1955, nowabandoned, which is a continuation-in-part of my copending applicationsSerial Number 480,033, filed January 5, 1955, now Patent No. 2,999,322,Serial Number 500,325, filed April 11, 1955, now Patent No. 3,101,645,and Serial Number 503,211, filed April 22, 1955, now Patent No.2,975,670. In these copending applications and in copending applicationSerial Number 548,842, filed on even date herewith, now Patent No.3,015,988, and entitled Perspective Alteration Means and Method I haveshown various methods and means by which images having the appearance ofplane areas as viewed from particular viewpoints may be altered toprovide images having the appearance of the same areas as viewed fromdifferent angles, or at displaced viewpoints. Method and apparatuscapable of such image alteration is of considerable use in numerousapplications, including, for example, apparatus for producing realisticvisual displays for use in grounded training equipment, apparatus forslanting lettering, designs and drawings to produce unique effects, andapparatus for use in conjunction with camera or film printer equipmentto provide film having the appearance of having been taken from a remoteor inaccessible location.

My copending application Serial Number 500,325, filed April 11, 1955,for Simulated Viewpoint Displacement Method and Apparatus, now PatentNo. 3,101,645, illustrates in detail method and apparatus for alteringthe apparent perspective of images by varying the images anamorphicallydifferent amounts in two perpendicular directions, and a preferredembodiment of the invention of the above-mentioned copending applicationshows ap paratus comprising a pair of perpendicularly operating variablepower anamorphosers. Since fixed power anamorphosers may be constructedat less cost and optical assemblies using fixed power anamorphosers maybe constructed at less cost and operated by simpler mechanicalapparatus, it becomes desirable to provide image alteration method andapparatus utilizing constant anamorphic magnification as much aspossible in lieu of variable anarnorphic magnification. Furthermore,while the abovementioned copending application shows a system in whichboth anamorphic image alteration means are rotatable as a unit about thesystem optical axis, it becomes desirable to provide comparable methodand apparatus in which the pair of anamorphic means are independentlyrotatable. Generally speaking, one who builds perspective alterationapparatus finds particular systems advantageous in particularapplications, and the availability of number of differing systemsgreatly facilitates the design of a commercially desirable product.

I have discovered that by providing two primitive 3,240,120 PatentedMar. 15, 1966 ice transformations of an image with selected powers andin selected directions, that the apparent perspective of the image maybe altered. I have developed a plurality of systems in accordance withthat discovery, so that the benefits of being able to maintain the poweror angle of a particular primitive transformation means always at thesame value regardless of desired change in perspective may be utilized.

It is therefore a primary object of the invention to pro vide apparatusfor altering the apparent perspective of planar images by means of twoprimitive transformation devices.

It is a further object of the invention to provide apparatus of theabove nature in which either the power or the direction of any one ofthe primitive transformation devices is maintained constant at a desiredvalue.

It is an additional object of the invention to provide apparatus of theabove nature in which the power of either of the primitivetransformation devices is maintained constant at a desired value.

Other objects of the invention will in part be obvious and will in partappear hereinafter.

The invention accordingly comprises the several steps and the relationof one or more of such steps with respect to each of the others, and theapparatus embodying features of construction, combinations of elementsand arrangement of parts which are adapted to effect such steps, all asexemplified in the following detailed disclosure, and the scope of theinvention will be indicated in the claims.

For a fuller understanding of the nature and objects of the inventionreference should be had to the following detailed description taken inconnection with the accompanying drawings, in which:

FIG. 1 shows in perspective with certain parts partially cut away aspecific optical embodiment of the invention utilizing one fixed poweranamorphoser and one variable power anamorphoser. In the specificationthe apparatus of FIG. 1 is designated as a Type -III system.

FIG. 2 shows in perspective with certain parts partially cut away analternative optical embodiment of the invention using one fixed poweranamorphoser and one variable power anamorphoser. In the specificationthe apparatus of FIG. 2 is designated as a Type IV system.

FIG. 3 shows in perspective with certain parts partially cut away analternative optical embodiment of the invention utilizing a pair ofvariable power anamorphosers. In the specification the apparatus of FIG.3 is designated as a Type H system.

FIGS. 4a through 4:! are geometrical diagrams useful in understandingthe nature of perspective alteration or viewpoint displacement of animage.

FIGS. 5a and 5b are gometrical diagrams useful in understanding thecapability of two primitive transforrna tions such as those provided 'bytwo anamorphosers to alter the apparent perspective of an image.

FIG. 6 shows in perspective with certain parts partially cut away aspecific optical embodiment of the invention utilizing two variablepower anamorphosers, each of which are independently variable in powerbut which have their axes of variable magnification aligned at a certainfixed angle with respect to each other. In the specification theapparatus of FIG. 6 is designated as a.

Type I system. Reference may also be had to my copending applicationSerial Number 500,325 which illustrates a particular species of Type Isystem.

FIG. 7 shows in perspective with certain parts partially cut away anoptical embodiment of the invention utilizing two anamorphosers eachvariable both in power and in angular orientation, which are controlledin accordance with an arbitrary restrain. In the specification theapparatus of FIG. 7 is designated as a Type V system.

FIG. 8 is an electrical schematic diagram of an exemplary controllerwhich may be utilized to operate a Type I system of the invention suchas shown in FIG. 6.

FIG. 9 is an electro-mechanical schematic diagram illustratingtelevision apparatus which may be used in practicing the method of theinvention electrically rather than optically.

FIG. 10 is an electrical schematic diagram illustrating the exemplaryapparatus which may be utilized to receive input quantities commensuratewith desired perspective alteration of an image and to provide outputpotentials for use in controlling apparatus shown in FIGS. 11 through16.

FIG. 11 is an electrical-schematic diagram of an exemplary computercontroller which may be used to operate a Type I system of the inventionsuch as is shown in FIG. 6.

FIG. 12 is an electrical schematic diagram of an exemplary computercontroller which may be used to operate a Type II system of theinvention such as is shown in FIG. 3.

FIG. 13 is an electrical schematic diagram of an exemplary computercontroller which may be used to operate a Type III system of theinvention such as is shown in FIG. 1.

FIG. 14 is an electrical schematic diagram of an exemplary computercontroller which may be used to operate a Type IV system of theinvention such as is shown in FIG. 2.

FIG. 15 is an electrical schematic diagram of an exemplary controlcomputer which may be used to operate a Type V system of the inventionsuch as is shown in FIG. 7.

FIG. 16 is an electrical schematic diagram of a controller which may beutilized in conjunction with any of the other controllers shown toprovide a counter-rotation so as to maintain line-at-infinity portionsof perspectively altered images parallel.

Shown in heavy lines in FIG. 4a is a trapezoidal or keystone-shaped areaABCD such as the appearance a rectangular surface might present whenviewed in perspective at a point situated at a particular place in linewith the centerline YY of the surface. From a position higher inaltitude than the initial viewpoint, the area might have an appearancesuch as trapezoid A'B'C'D', and when viewed from a position lower inaltitude than the initial viewpoint, the area might have an appearancesuch as trapezoid AB"C"D". In FIG. 4:: line HH represents the horizon orline at infinity. Shown in FIG. 4b is a side elevation view showing aneye situated at point P viewing a rectangular surface at an altitude itabove said surface, the side BC of said surface being shown as a heavyline. It will be seen that if a screen S is placed a distance q in frontof viewpoint P, that a replica of the actual scene viewed from viewpointP may be simulated by presentation of a proper scene on screen S forobservation from point P. Assuming that screen S is mounted in agenerally vertical position as shown, it may be seen that in order toeffectuate a realistic presentation, that the distances of objects belowthe horizon line on screen S must be inversely proportional to theactual horizontal distance between these points and the ground positionof the viewpoint. For example, the distance I1 on screen S between thehorizon and the simulated near end AB of the surface must be inverselyproportional to R the horizontal distance between view point P and theactual near end 4 AB of the grounded surface, or as may be seen bysimilar triangles:

h =h Similarly, the distance 11 on screen S between the horizon and thesimulated far end CD of the rectangular surface is inverselyproportional to R the horizontal dis tance between viewpoint P and theactual distance to the far end of the rectangular surface, or that:

It may now be appreciated that for proper presentation of a scenesimulating a surface seen in perspective, that increases in viewpointaltitude require proportionate increases in distances h and 11 of such ascene, and that conversely, decreases in viewpoint altitude requireproportionate decreases in distances I1 and I1 of such a scene. Hence ifa photograph were taken of a scene at a particular viewpoint, anappropriate stretching or squeezing of the image from such photographwith respect to the horizon would yield scenes such as those viewed atpoints above and below the point where the picture was taken in the sameplane as that in which the photograph was taken.

Shown in FIG. 40 are the appearances which a rectangular surface mighthave when viewed from three viewpoints of the same altitude but varyingin lateral position with respect to the surface. The center portion ofFIG. 40 illustrates the scene which might be viewed from a viewpointlocated on the longitudinal centerline of the surface. The left-handportion of FIG. 40 illustrates the same surface viewed from a locationlocated a distance a to the right of the centerline of the surface, andthe right-hand portion of FIG. 40 illustrates the same surface viewedfrom a viewpoint located a distance b to the left of the centerline ofthe surface. Superimposed upon each portion of FIG. 4c in dashed linesis a rectangle which may represent a photographic slide which might beused to project a simulated scene. It may be seen that the displacementsa and b of the centerline on the slide at the lower edge of the frameare proportional to the ratio of the lateral displacement of theviewpoint to altitude of the viewpoint. If pictures were taken so thatthe horizon in each picture would be located along the upper edge of theframe, then the lateral displacement of any point in the picture fromits position in the center portion of FIG. 40 is proportional to thedistance from the point to the top of the frame. Thus it may be seenthat by providing distortion of an image varying in ac cordance with themagnitude of lateral viewpoint displacement from a reference viewpointand varying linearly from zero distortion at the line at infinity orhorizon to maximum distortion at a nearest location, that scenes varyingin accordance with lateral displacement of a viewpoint may be produced.I have designated such distortion as shear distortion since it producesa shape similar to those produced by applying shear forces to an elasticmember. Now it should be understood that by stretching or squeezing animage of an area with respect to its line at infinity or horizon, and byshearing the image linearly as described above, images may be altered toprovide resulting images which have a different center of perspectivelocated in the plane of the original viewpoint. The plane of theoriginal viewpoint may be seen to be the plane passing through theviewpoint which is perpendicular in two directions from the line-0fsightof the original viewpoint. The distortion required to simulate viewpointdisplacement is explained in a slightly different manner in my copendingapplication Serial No. 511,488. Although the above explanation is givenprincipally in terms of an outdoor scene in which the line at infinityis the actual horizon, it should be noted that the theory applies quiteas readily to all other images of perspective views, and the termsvertical and horizontal are therefore used broadly.

Assume that the rectangle of FIG. 4d represents an image of a surface asviewed from an original viewpoint, such as the image which might beformed by photographing the surface from the original viewpoint. Let theupper edge of the rectangle represent the horizon or line at infinity ofthe image. If the original image is compressed or squeezed and shearedin accordance with the rules given above for a viewpoint displacement,it will result in a parallelogram image having a new height and a slope,perhaps as shown by the parallelogram of FIG. 4d. For producingdistortion to simulate a given viewpoint displacement, threerelationships between the undistorted image (rectangle) and thedistorted image (parallelogram) may be determined: (1) the ratio ofheights, 11 to h; (2) the slope angle a; and (3) the fact that thehorizon dimension in FIG. 4d) remains constant.

Assume that an image of an area or a portion of an image of an area isrepresented by the rectangle of FIG. 5a. As shown in the figure, therectangle has a height of I1 and a width C. It will be apparent from theexplanation given above in connection with FIG. 4, that verticalexpansion of the image to a new height h and shearing of the imagelaterally in an amount proportional to the distance d would provide animage having the shape of the parallelogram of FIG. 5a, and that suchparallelogram image would represent the original area as viewed from anew viewpoint. If the upper line of the rectangle of FIG. 5a is assumedto represent a line at infinity or horizon of the original image, itwill be understood that the upper line of the parallelogram shouldcoincide with the upper line of the rectangle, since things viewed atinfinity do not appear in different locations nor with different sizeswhen viewed from various viewpoints.

FIG. 5b shows the original rectangular image portion and alsoparallelogram images which may be produced by means of two co-axialanamorphosers acting on the original image. In FIG. 5b point 0represents the axis of an optical system containing a pair ofanamorphosers. The first anamorphoser is rotated about the optical axis0 so that its direction of anamorphic power (m axis in FIG. 5b) is at anangle 5 from a reference axis Y-Y ofv the original image, and theanamorphoser may have unity power in a direction perpendicular to the maxis, or along m in FIG. 5b. The effect of the first anamorphoser willbe to produce a parallelogram image such as that shown in dotted linesin FIG. 5b. The parallelogram may be constructed from the originalrectangle by lengthening the rectangle in all dimensions parallel to them axis by an amount proportional to the power of the first anamorphoserwhile maintaining all dimensions perpendicular to the m axis at theiroriginal length. In terms of analytic geometry, such alteration of afigure is termed a primitive transformation. The second anamorphoser isrotated about the system axis 0 so that its direction of anamorphicpower (m axis of FIG. 5b) is at an angle 0 from the m axis. The effectof the second anamorphoser will be to act on the image represented bythe intermediate parallelogram shown in dotted lines to provide an imagesuch as the resultant parallelogram shown in dashed lines. The resultantparallelogram may be constructed from the intermediate parallelogram byvarying each dimension of the intermediate parallelogram along the maxis by the power of the second anamorphoser, while maintainingdimensions perpendicular to the m axis (or along the m axis) at the samelength, or, in effect, making a second primitive transformation. It willbe apparent that the size and shape of the resultant parallelogram willdepend upon the powers of each anamorphoser and their angularorientations. While the upper lines of the original rectangle and theresultant parallelogram have not been shown as being of the same length,it will be apparent that if correct powers and angular orientations wereselected, that those dimensions could be made to be of the same length,and in analysis of FIG.

5b such equality will be assumed. If it be assumed that the original orrectangle image portion is produced by means of a film frame or similarobject, axis YY of FIG. 5b may represent, for example, the verticaldirection in the projected image, and axis XX may represent thehorizontal direction.

The effect of the two anamorphosers on the original rectangle imageprovides a rotation of the image as well as shearing and compression. Asshown in FIG. 5b the amount of such rotation is indicated by the anglep. It will be seen that if one wishes to maintain infinite distance orhorizon line portions of images in a fixed location on a screen or othersurface, that the resultant trapezoidal or distorted image must berotated through the angle p to compensate for the rotation which occursas an incident to expansion and shearing. If the image to be altered inperspective is to be cast upon a fixed screen, the object utilized toproduce the original image may be counter-rotated through an angle equalto p, or in systems utilizing a rotatable screen or other surface, thescreen may be rotated relative to the object. The former system isdeemed preferable for grounded training visual displays, but the lattersystem is quite acceptable for photographic slanting and other useswhere continuous projection is not necessary and where actualorientation in space is unimportant. In systems not requiring a largeangular field, the required counterrotation of the image may be effectedby means of rotatable Dove prisms, in a manner which will be apparent tothose skilled in the art.

It will become apparent from inspection of FIG. 5b that mere rotation ofthe resultant parallelogram through the angle p will not cause the upperlines of the parallelogram and rectangle to coincide, but that theparallelogram would also have to be shifted vertically and laterally.The amount of shifting required depends upon the distance betweenoptical axis 0 and the portions of the original image representing thehorizon or line at infinity. It will be apparent to those skilled in theart that if the horizon or infinite distance portion of the image wereto lie on optical axis 0, that such portion would not be displaced fromthe axis in the resultant parallelogram image, since axial light raysremain undeviatcd in a coaxial lens system. Therefore, if the originalimage is produced, for example, from a film made with the camera axispointed toward the horizon or line at infinity, and such film is thenprojected with its axiallytaken portion projected axially, no verticalor lateral shifting is required to maintain horizon portions coincidentas viewpoint displacement is varied. In using the invention forproduction of a training visual display, it is often desirable tomaintain the camera axis pointed away from the horizon in order toutilize more economically the available angular field, and method andmeans for providing corrective shiftings then required are described indetail in my copending application Serial Number 503,211 filed April 22,1955 for Method and Apparatus for Producing Visual Display, and theinvention of that application may be used in conjunction with thepresent invention.

To alter an image perspective from one viewpoint to an image perspectivefrom a desired viewpoint, three things may be determined as explained inconnection with FIG. 4; namely, (1) the ratio of the height h of thedesired image to the height h of the original image, (2) the sloping ofthe original image required to produce the desired image, which slopingis shown as the angle a in FIG. 5, and (3) the fact that horizon line orvanishing point portions of both images are the same, or in other wordsthat the upper lines of the rectangle and final parallelogram are ofequal length. Knowing the above three facts, a number of independentrelationships concerning FIG. 5b may be written. Since each of therelationships may be derived by means of elementary geometry,trigonometry and algebra, their derivations need not be set forthherein.

It will be seen that the above nine simultaneous equa tions contain nineunknowns (m m ,5, 0, 'y 6,56 and p) and three independent variable inputterms (d, 11 and I1 dependent upon the perspective alterationdesired.The unknown quantities m m ,3 and may each be termed a control variable,since each relates to an adjustment which may be made to one of theanamorphosers used in the invention. Unknown angle quantities 'y 6,, and6 are, of course, related to the four control variables in the mannerexpressed in the equations, but these unknowns do not in themselvesexpress directly and conveniently any physical adjustment which may bemade to either or both of the anamorphosers to provide a desiredviewpoint displacement. If desired, the nine simultaneous equationsgiven above may be solved analytically and simultaneously to eliminatethese four unknown angle quantities, providing four equations whichexpress the control variables in terms of the viewpoint displacementinputs d, 11 and 12 The required counterrotation angle p necessary tomaintain horizon lines parallel may be separated from the equationsneeded to specify the dependent control variables In m ,8 and 0,although the angle p itself is a function of those variables. The aboveequations presume that the original and the altered images are to be ofthe same angular field. As a matter of fact, the original image mayconsist, for example, of a film taken with a camera of a given focallength, while the projection system utilized in producing an alteredimage may be of quite a different focal length. In some uses of theinvention it may be considered desirable to utilize a conventional wideangle attachment. To produce an image having the same angular field asthe original image, any spherical magnification introduced into thesystem by a difference between camera and projector focal lengths, orany spherical magnification introduced into the system by use of a wideangle attachment or like device should be accounted for in theequations. If such spherical magnification is designated P theright-hand sides of Equations (5) and (6) should be multiplied by P asfollows:

Z Z= -P m m i i- (5 COS=P HLflH 22 7 afi (6a) where and wherein=efifective focal length of camera lens used to provide original imageon film, including wide angle attachments, etc., if any;

,=efiective focal length of the projection lens system (exclusive of anyeffects produced by the anamorphosers), including wide angleattachments, etc., if any d =projections throw or distance d =viewingdistance The constant term P shown as the product of system sphericalangular magnification and the ratio between projection distance toviewing distance is designated system spherical magnification forconvenience. Although FIG. 4 illustrates an arrangement in which theprojection system is coincident with the viewpoint, it will be apparentthat in actual practice of the invention, a projector may be displacedtherefrom provided an adjustment of focal lengths is made in accordancewith the above expression.

Since the relation between an undistorted and a distorted image requiresthe specification of three facts, three of the control variables of thesystems of the invention must be varied to provide continuous viewpointdisplacement. Therefore, provision of two anamorphosers, which togetherhave four controllable variables, requires that one of the controllablevariables be maintained constant, or, if allowed to vary, that anadditional rsetraint be imposed upon the system. The invention thereforeembraces several basic types of differentlyarranged apparatus, all ofwhich operate in accordance with the relationships expressed above, andwhich types may be tabulated as follows:

Type Anamorphoscr settings maintained constant V m 7712, B, 9

i, 6 m 'lltg, 0 nu. B, 0. 111 B, 19.

Ill]. None, but additional restraint imposed.

As well as the systems tabulated herein, I have discovered that a numberof further systems may be constructed in which the system sphericalmagnification P is varied. Such further systems are shown, described indetail and claimed in my copending application, now Pat. 3,015,988,filed on even date herewith, and entitled Perspective Alteration Meansand Method.

As will be apparent to those skilled in the analogue computer art, theabove eight simultaneous equations may be solved simultaneously byprovision of eight interconnected servos, or the number of simultaneousequations may be reduced by analytical simultaneous solution to as fewas four simultaneous equations containing the four control variableunknowns (m m 5 and 0), and such four simultaneous equations could besolved in similar manner by three interconnected servos if an arbitraryrestraint is placed upon one of the unknowns. Since the dynamic behaviorof systems having a large number of interconnected servos is usually notreadily analyzed, making stabilization of such systems sometimes quiteditiicult, I usually prefer to solve analytically and explicitly for thecontrol variables themselves or for simplified functions of the controlvariables, and to utilize servo sys' tems solving at least in part forthe control variables themselves from the input data, so that the numberof interconnecting loops is considerably lessened. Shown, however, inFIG. 8 is an electrical schematic diagram of a computer controlconnected to provide the proper outputs for operating Type I apparatusof the kind shown in FIG. 6 (wherein m m and pt are variable and 0 ismaintained constant), in which computer control the eight equationsgiven above are solved simultaneously.

System Type I is designated above as having m m and 13 as variable, with0 fixed at a constant value, or in other words, that the powers of bothanamorphosers and the angular orientation of the first anamorphoser arevaried as viewpoint displacement varies, but that the angularorientation of the second anamorphoser with respect to the firstanamorphoser is maintained constant. Reference to my copendingapplication Serial Number 500,325 will show that the optical systemdisclosed therein is a special case of system Type I, in which is fixedat a specific value of ninety degrees. In constructing a Type I systemof the invention it is not necessary that 0, the angle between the axesof the anamorphosers be maintained fixed at ninety degrees, and in FIG.6 there is shown a physical arrangement in which 0 has been fixed atapproximately 45 degrees.

System Type II utilizes m m and 0 as variables and maintains 13constant, allowing construction of a system in which the firstanamorphoser need not be rotated with respect to the original image. Insystems utilizing a film or slide projector, this feature allows rigidmounting of the first anamorphoser to the projector case. An exemplaryarrangement of apparatus of system Type II is shown in FIG. 3. SystemType III utilizes m 6 and 0 as variables and maintains m constant,allowing construction of projection system in which the outermostanamorphoser need not be varied in power. An exemplary arrangement ofapparatus of system Type III is shown in FIG. 3. System Type IV utilizesm {3 and 0 as variables and maintains m the power of the firstanamorphoser, constant, thereby simplifying certain mechanicalarrangements. An exemplary arrangement of system Type IV is shown inFIG. 2. As mentioned above, system Type V varies the powers and angularorienta tions of both anamorphosers, but impose an additional restraintupon the system. If the particular restraint selected is (fi+0)=aconstant, it will be seen that the second anamorphoser, the angularorientation of which corresponds to (5+6) with respect to the originalimage may be fixed in relation to the projector or other image producingapparatus, and hence pinion 118 and its driving means (not shown) shouldbe eliminated. It will be apparent that a practically infinite number ofarbitrary restraints exist, and selection of the most useful restraintdepends to a great extent upon the allowable complication of themechanical apparatus and the control computer utilized. If desired, thearbitrary restraint may consist of maintaining the requiredcounter-rotation angle p equal to a constant, so that the images neednot be axially rotated to keep horizon lines coincident. The method andmeans by which such arbitrary restraints may be imposed on the controlsystem will be further described below, and an exemplary control systemfor a system involving an added arbitrary restraint will be illustrated.

Referring to FIGS. 1, 2, 3, 6, and 7, there are shown exemplarymechanical arrangements for several different optical systemsconstructed in accordance with the invention, and in these figures likenumerals correspond to like parts. It should be emphasized that themechanical arrangements shown for varying anamorphoser power and foraxially rotating the anamorphosers are illustrative only. While I haveshown these various systems as attachments uitable for addition to aprojector, it will be immediately apparent that the various opticalelements may be mounted and housed in many different arrangements. Itwill also be apparent that while I have shown the optical elements ofeach system coaxially arranged physically, that reflectors and similarapparatu may be used in the systems of the invention, and it isnecessary only that the optical elements be coaxial optically.

FIG. 1 shows an image-producing source PR such as a conventional slideor motion picture projector, which casts an image toward the right asviewed in the figure along optical axis 0-0. The image maybe focused ona screen (not shown) or other surface. In utilizing the invention forproduction of slanted lettering or perspectively-altered stillphotographs, the image may be focused onto a photo-sensitive surface.Mounted for axial rotation on projector PR is a lens barrel 101. Toothedflange 102 of barrel 101 is journalled in the projector housing, and theends of barrel 101 are rotatably supported by hearing pedestals 103 and104. Pinion 105 engages toothed flange 102, so that rotation of pinion10S serves to rotate lens barrel 101. Pinion 105 may be rotated by a ,8servo-motor M-300 (not shown). Carried within lens barrel 101 are twopositive cylindrical lenses, L and L and a negative cylindrical lens Lwhich three cylindrical lenses have their axes of magnification alignedso as to form a variable power anamorphoser. Positive cylindrical lensesL and L each are slidably mounted in barrel 101 by means of longitudinalkeyways (not shown), which constrain the lenses from axial rotation withrespect to each other. Cam pins 106 and 107 are rigidly attached tolenses L and L respectively, and

protrude through longitudinal slots cut in lens barrel 101.

Cam pins 106 and 107 also protrude through non-linear cam slots such as108 cut in a rotatable sleeve 109 which surrounds barrel 101. An mservo-motor M-100 is rigidly mounted to barrel 101 by means of mounting110, and pinion 111 on the shaft of motor M-100 drives a toothed flangeportion 112 of rotatable sleeve 109, thereby rotating sleeve 109axially. As sleeve 109 rotates,.the non-linear cam slots move positivecylindrical lenses L and L axially with respect to fixed negativecylindrical lens L changing the anamorphic magnification of the firstanamorphoser. The relationships between lens movement and magnificationof such type variable anamorphoser are shown and explained in detail inmy copending application Serial Number 480,033 and need not be repeatedherein. Furthermore, other types of variable anamorphosers may besubstituted for the type shown without departing from the invention.Mounted co-axially with the first anamorphoser described above is asecond anamorphoser comprising negative and positive cylindrical lensesL and L each of which are fixedly mounted within a rotatable lens barrel115, which is r0- tatably supoprted by bearing pedestals 116, 117.Rotation of pinion 119 through the angle 0 drives toothed flange portion118 of barrel 115, thereby axially rotating the second anamorphoser.Lenser L and L have their axes of magnification aligned with each other,but since the distance between lenses is not varied, the secondanamorphoser is not variable in power. While I have shown the apparatusof various embodiments of the invention as being motor-positioned, itwill be apparent that in many uses of the invention, as, for example,where alteration of still images is desired, that the optical elementsmay be manually-positioned, and suitable dials and scales may beprovided on the apparatus to facilitate setting the various elements todesired powers and angles.

FIG. 2 shows a system having a rotatable, fixed power first anamorphoserand a rotatable variable power second anamorphoser. An m servo-motorM-200 varies the power of the second anamorphoser of FIG. 2 in the samemanner mechanically as the m servo of FIG. 1 varies the firstanamorphoser. Servomotor M-300 (not shown) angularly positions the fixedpower first anamorphoser by means of pinion 105, and rotation of pinion119 angularly positions the second anamorphoser. FIG. 3 shows a systemin which the first anamorphoser is variable in power but not rotatable.Pinion 111 is positioned by m servomotor M-100 (not shown), and lensbarrel 101 may be rigidly aflixed to the projector case. Motor M-400rotates lens barrel through the angle 0, and motor M-200 varies thepower of the second anamorphoser in the same manner as in FIG. 2.

FIG. 6 illustrates system Type I apparatus, in which both anamorphosersare mounted within the same lens barrel 101a, so that the angle 0between their axes of magnification is maintained constant. In FIG. 6the angle 0 is shown at approximately 45 degrees. Lens barrel 1014 isaxially rotated through the angle ,3 by means of servo M-300 (not shown)which is geared to toothed flange 102 by barrel 101a by means of pinion105. Servo M-100 rotates sleeve 109 by means of pinion 111, so thatcurved slots in sleeve 109 may operate on cam pins 106 and 107, therebyaxially moving lenses L and L with respect to lens L Similarly, servoM-200 varies the power of the second anamorphoser, cam pins 122 and 123serving to position lenses L and L with respect to fixed negativecylindrical lens L As mentioned above, a specific embodiment of Type Iapparatus in which the two anamorphosers are maintained perpendicular asshown in my copending application Serial Number 500,- 325. It is notnecessary in constructing Type I apparatus that the two anamorphosers bearranged at 90 degrees or at any multiple or submultiple of 90 degrees.It is necessary only that the axes of magnification of the twoanamorphosers not be aligned at zero or 180 degrees if lateraldisplacement of the viewpoint is to be provided in the perspectivelyaltered image.

FIG. 7 illustrates a Type V embodiment of the invention, in which allfour of the control variables are varied, so that an arbitrary restraintmust be improved on the control computer. Since parts of the apparatusof FIG. 7 are numbered similarly to parts of the preceding figures, adetailed description of FIG. 7 is deemed unnecessary.

It will be recalled from FIG. 5 that 0 represents the angle between thepower axes of the first and second anamorphosers, and hence if the firstanamorphoser is rotated through the angle #3 with respect to the imageprojector, any 0 servomotor used to angularly position the secondanamorphoser should be either carried bodily with the first anamorphoserso that 0 will be measured from the m axis, or else the secondanamorphoser should be rotated through the sum of the 6 and 0 angles bya shaft fixed with respect to the image projector. Hence if the precisemechanical arrangement of FIG. 1 were used, pinion 119 should bepositioned in accordance with {3 and 0 either by controlling servo M460in accordance with (3+6), or by driving pinion 119 from a mechanicaldifferential having B and 0 inputs from servos M-300 and M-400. The sametheory applies to each of the other arrangements should, except, ofcourse, in FIG. 6 where no 0 rotation is necessary.

While I have shown specific types of variable power anamorphoscrs inillustrating the invention, other types may be readily substituted, asfor example, prism-type variable anamorphosers such as the Hi-Lux Valtype, made by Projection Optics Co. of Rochester, NY. and theSuper-Panatar and Ultra-Panatar types manufactured by RadiantManufacturing Coporation of Chicago, Illinois. The optical elements ofboth anamorphosers may be interleaved in some embodiments of theinvention, although this usually leads to greater mechanicalcomplication of the system. I have shown each anamorphoser as anattachment which may be added to a conventional projection system, butit will be readily apparent to those skilled in the art that ordinaryspherical projection lenses may be carried in the variable anamorphoserhousings rather than on the image projector, and in some uses of theinvention, no projection lenses would be required. Furthermore, wideangle attachments and/or fixed-power non-rotatable anamorphicattachments may be added to the systems to obtain wider field coverageand/or the usual benefits of Cinemascope type projection withoutdeparting from the invention.

In FIG. 8 a servo is provided for each of the known variables of theeight equations, and an input means is provided for setting 0, the anglebetween the first and sec ond anamorphoser axes, at a desired value.Also provided is a spherical magnification input control to be set inaccordance with any spherical magnification introduced into the systemby means of a wide angle attachment or like device, or by use of acamera and projector combination having different focal lengths. Inorder that conventional analogue computer sin-cosine resolvers may beused, the tangent terms of the equations have been replaced by sin/costerms, and the equations have been multiplied to eliminate fractions,thereby simplifying the computer. In FIG. 8 and other electricalschematics herein sine and cosine resolvers have been shown as simplepotentiometers for sake of clarity and convenience of explanation, andthose skilled in the art will recognize that resolvers capable of 360degree resolution may actually be used. The feedback amplifiers shownmay comprise conventional analogue computer summing amplifiers, havinghigh loop gain and unity overall gain, although series summing may beused if desired. A number of amplifiers used for polarity inversion anda number of butter amplifiers have been omitted for sake of clarity. Theservos shown in block form may comprise completely conventional analoguecomputer servos, employing conventional amplifying means, appropriateelectrical, mechanical or hydraulic motive means, and such servos may beprovided with numerous well-known refinements and constructions]delails, such as tachometer generator or other rate feedback, reductiongearing, mechanical limit stops, etc. While I have shown the 0 and Pinput controls as being manually adjustable, so that the controller maybe used with systems having various values of 0 (second anamorphoserangle) and P (spherical magnification), it should be understood that inconstructing a controller for use with a system using specific values of0 and P that the potentiometers and resolvers shown as manuallyadjustable may be replaced by fixed resistors. it a value of 0 isjudiciously selected, the equations will greatly simplify. Reference maybe had to my copending application Serial Number 500,325 wherein 0 isset at degrees, so that sin 0 becomes unity and cos 9 becomes zero, anda simplified controller has been constructed using the simplifiedequations.

The v servo M-l solves Equation (1) in its modified form:

sin [3 cos m sin 7 cos 3:0 (lb) A potential proportional in magnitude tothe first term of Equation (lb) is derived by means of sine resolver RII and cosine resolver R-12, and is applied via summing resistance R-13to the input circuit of servo M-l. The second term of Equation (lb) isderived by means of linear potentiometer R-16, the arm of which ispositioned by m; servo M-5, sine resolver R-15, and cosine resolverR-14, and the potential proportional to such term is applied via summingresistance R-17 to the input circuit of servo M-l, to be summed with thepreviously mentioned potential. Each of the other servos receives inputpotentials in a similar manner. The 6, servo M-2 solves Equation (2) inthe following modified form:

sin 6 sin ('y,. +6)+m cos 6,, cos (y -H) =0 (2b) The first term ofExpression (2b) is derived by means of sine resolvers R-18 and R-19. Theshaft of resolver R-l9 may be seen to be positioned in accordance withthe angle (n+0) by the output of differential 801, which receives shaftinputs commensurate with the angle 7,, from servo M1 and the angle 0from manual input control knob 802. The second term of Expression (2b)may be seen to be derived by potentiometer R-20, cosine resolvers R-21and R-22. The potentials proportional to the first and second terms areapplied to the input circuit of servo M-2 via summing resistors R-23 andR-24, respectively.

The a servo M-3 solves Expression (3) in the following modified form:

sin cos ;3m sin 5 cos =0 (3b) A potential proportional to the first termof Expression (3b) is derived by means of sine resolver R-25 and cosineresolver R26, and is applied to the input circuit of servo M-3 viasumming resistor R-27. A potential proportional to the second term ofExpression (3b) is derived by means of potentiometer R-28, sine resolverR-29 and cosine resolver R-30, and is applied to the input circuit ofservo M-3 via summing resistor R-31.

The m servo M-4 solves Expression (4) in the following modified form:

m sin 6 sin cos (n+0) cos 6 =0 (4b) 11 cos ,8 cos (v -H) +m m h P sin 8cos -y,,=0 (b) A potential proportional to the altitude 11 of thedesired viewpoint is applied at terminal 804, modified by cosineresolvers R-37 and R-38, and applied via summing resistor R-39. Thesecond term potential of Expression (5b) is derived by means ofmanually-adjusted P linear potentiometer R40, linear potentiometers R-41and R-42, sine resolver R-43 and cosine resolver R-44, and the modifiedpotential is applied to the input circuit of servo M-5 via summingresistor R-52. Manually adjusted potentiometer R-40 may be adjusted inaccordance with any spherical magnification introduced into the systemas mentioned above.

The p servo M-6 solves Expression (6) in the following modified form:

cos ix sin ,6 sin (n+0) -!7Z I7I Pn cos 5,, sin 7 :0 (6b) The potentialproportional to the first term of Expression (6b) is derived by means ofcosine resolver R-46 and sine resolvers R-47 and R-19, and is appliedvia summing resistor R-48 to the input circuit of servo M-6. The secondterm potential is derived by means of linear potentiometers R-Sl, R-52,R-53, sine resolver R-49 and cosine resolver R-50, and is applied viasumming resistor R-54 to the input circuit of servo M-6.

The a servo M-7 solves Expression (7) in the following modified form:

11 sin a-d cos a=0 The potential proportional to the first term ofExpression (7b) is derived by modifying the 11 input potential fromterminal 804 in accordance with sin a by means of sine resolver R-55,and this potential is applied to the input circuit of servo M-7 viasumming resistor R-56. The second term potential is derived by applyingthe lateral displacement or d input potential to cosine resolver R58from terminal 805.

The 5 servo M-8 solves Expression (8) in the following modified form:

its input signal, and therefore, that as the I11, I1 and d inputpotentials are varied, the servos will reposition themselves so as tomaintain their shafts at positions commensurate with their respectivevariables. Servo M-5 may be mechanically connected via pinion 111 to thefirst anamorphoser of FIG. 6 to vary m the power of the firstanamorphoser. Servo M-4 may vary the power m of the second anamorphoservia pinion 121, and p servo M-6 may axially rotate the image alterationapparatus with respect to the original image by means of pinion 105.While I have shown a specific computer for controlling the apparatus ofFIG. 6, it is not at all necessary that the particular servos beutilized to solve for the particular variables. Those skilled in the artwill recognize as a result of the disclosure, that the relationshipsbetween an original image and an altered image defined by the equationsgiven may be altered in an infinite number of different ways withoutdeparting from the invention.

By solving equations (1) through (9) analytically and simultaneously,one may provide further equations which are more convenientlymechanized. By elementary algebra and trigonometry the followingequations may be obtained:

In the above solution the system spherical magnification P has beenincluded by using Equations (5a) and (6a). If a spherical magnificationof unity is used, P may be replaced by 1.0 in the equations. The abovefour equations may be re-arranged and solved together in very many ways,to provide further equations which may be deemed preferable for analogcomputer solution in par ticular embodiments of the invention.

Referring now to FIG. 10 there is shown an exemplary form of a portionof a control computer. The independent variables which express desiredviewpoint displacement are shown as being applied to manuallypositionable control knobs 1000, 1001 and 1002. In certain embodimentsof the invention the shafts shown as positioned by the aforesaid controlknobs may be set automatically as by means of servos, for example. In mycopending applications Serial Numbers 480,033 and 500,325 I have shownsystems in which such input quantities are automatically provided inaccordance with the instan taneous location of a simulated aircraft. Theapparatus of FIG. 10 receives the independent variable input data andprovides a number of output potentials which are various functions ofsaid variables. These output poten tials, which are present at theterminals at the right hand side of FIG. 10, may be used to actuate thevarious con trollers shown in FIGURES 11 through 16. Control knob 1000may be set to the altitude I1 (measured in the plane of the originalviewpoint) of the original viewpoint. Control knob 1001 may be set tothe altitude I1 (measured in the same plane) to which the resultantviewpoint is required to be located. Control knob 1002 may be set inaccordance with the desired lateral displacement of the desiredviewpoint measured in the same plane from the originial viewpoint.Potentiometer R-1001 is excited by a constant potential by the computerpower supply and its wiper arm is positioned by knob 1000 to provide aninput potential proportional to h via resistor R-1004 to amplifierU-1001. The h output potential from amplifier U-1001 is applied toterminal 1003, and also inverted in polarity or sign by amplifier U-1010and applied to terminal 1004. In similar manner, h and +11 potentialsare developed by potentiometer R-1005, amplifiers U-1002 and U-1012 andapplied to terminals 1008 and 1010, respectively.

The h output potential from amplifier U-1001 is applied to excitepotentiometer R-1002, the arm of which is positioned by knob 1000,deriving a --h potential which is applied to summing amplifier U-1006and which also is inverted in sign by amplifier U-1017 and applied tosumming amplifier U-1008 via summing resistor R4023. A -h potential isdeveloped in similar manner by potentiometer R-1006 and applied viasumming resistor R-1035 to amplifier U1006. A +d potential derived bypotentiometers R-1007 and R-1008 is inverted in sign by amplifier U-1007and applied via resistor.R-1036 to amplifier U-1006. The sum of thesepotentials from amplifier U-1006 modified in accordance with h bypotentiometer R-1010, further modified in accordance with 11 bypotentiometer R-1020, and fed back to the input circuit of amplifierU-1006, via resistor R-1037. As those skilled in the art willimmediately recognize, the modification of the summed input potentialsby 11 and 11 in the manner shown serves to provide an output potentialfrom amplifier U-1006 divided by (11 11 so that the potential applied toterminal 1006 is proportional to the quantity:

This potential is also inverted in sign by amplifier U-1011 and madeavailable on terminal 1005. In similar manner an output potentialproportional to is provided by amplifier U-1008 and applied to terminals1011 and 1012. The -h output potential from amplifier U-1001 applied toamplifier U-1005 is divided by I1 by means of potentiometer R-1015,providing an output potential proportional to h /h on terminal 1009.Similarly, the output potential --h from amplifier U-1002 is divided byIt; by means of amplifier U-1004 and potentiometer R-1003 and appliedwith opposite signs to terminals 1015 and 1007, and similarly, d/h and-d/h potentials are derived and applied with various polarities toterminals 1013, 1014 and 1016. Each of the amplifiers shown in FIG. 10may comprise a conventional analogue computer feedback amplifier.

The controllers for the various systems of the invention shown in FIGS.11 through 14 each are shown provided with a manually adjustable P orspherical magnification control and a manually adjustable control whichmay be set in accordance with the selected constant value of either thepower or the angular position of one of the anamorphosers. Thecontroller shown in FIG. 15 is provided solely with a sphericalmagnification control since both the powers and angular positionsof bothanamorphosers of a Type V system vary as perspective alteration isvaried. It should be understood that in providing controllers for usewith specific embodiments of the invention wherein system sphericalmagnification and/ or the other manually adjustable shaft need not bevaried, that the variable potentiometers and resolvers shown as beingadjusted by the manually operable shafts may be replaced by fixedresistors. Furthermore, the use of p or 0 angles of 0, 90, 180, and 270degrees in those systems in which ,8 or 0 are constants will be seen tosimplify greatly the equations, so that many terms of the equationsbecome zero or unity, and the equations may be mechanized using manyless parts. Certain limitations on the selection of these angles willbecome apparent either by examination of the equations, or preferablyfrom examination of plots of the equations. For example, in system TypeI wherein 0, the second anamorphoser angle, is maintained constant, thevalue of 0 must not be set at zero degrees (or degrees) if displacementof the viewpoint over an area is to be obtained, since bothanamorphosers would be acting along the same axis and could not controlimage height and image shearing independently. Similarly, the angle 5,the angle of the first anamorphoser must not be set to zero or 180degrees if a continuous displacement Type II system is to be provided.In systems Types III and IV wherein m or m the power of one of theanamorphosers is maintained constant, it will be apparent that suchpower should not be unity, since an anamorphoser having unity power isinoperative to affect the shape of the image. In the illustrated Type Vsystem wherein the powers of both anamorphosers are maintained equalalthough variable, it is necessary that system P not equal unity iflateral displacement near the original viewpoint is desired. Equations(10) through (13) may be plotted on lateral displacement a and resultantviewpoint altitude I1 coordinates by assuming an original viewpointlocation il a particular system spherical magnification P and aparticular value for the control quantity maintained constant, and bysubstituting various values of the three variables into the equations.The characteristics of a particular embodiment of the invention may beunderstood very easily from such charats. Charts of this nature arereproduced in my copending application Serial Number 548,842, filed oneven date, now Pat. No. 3,015,988, my copending application SerialNumber 500,325, now Pat. No. 3,l0l,645 and my copending applicationSerial Number 511,488, now Pat. No. 2,975,671, and reference may be hadto such applications.

Referring now to FIG. 11 there is shown in electrical schematic form anexemplary controller for operating a Type I embodiment of the inventionsuch as that illustrated in FIG. 6, wherein the powers of bothanamorphosers are varied, the angular position )3 of a firstanamorphoser with respect to the original image is varied, but theangular position 0 of the second anamorphoser with respect to the firstanamorphoser is maintained constant. Control knob 1101 may be adjustedin accordance with the value at which the angle 0 is maintained. The mservo shown in block form receives input potentials which are variousfunctions of the independent variables (h h d) and functions of theconstant quantities 0 and P the system spherical magnification. SolvingEquation (12) for m and substituting the answer into Expression (10),the following expression is obtained, and this expression is solved bythe m servo:

h 1 hP 1 2 2 l o hm. ht t. (14) '12 1 potential is divided by m} bypotentiometers R-1112 and R-1113 with amplifier U-1109 and applied toamplifier U-1105. The

potential is multiplied by m by means of potentiometers R-1109 andR-1110'and applied to amplifier U-1105. The output potential fromsumming amplifier U-1=105 may be seen to be proportional to thebracketed quantity in the above expression. This potential is multipliedby sin by means of resolvers R-1101 and R-1102 and the resultingpotential applied to the input circuit of the m servo via summingresistor R-1107. Since the potentials applied to the m servo willbalance to zero when and only when potentiometer R-1109, R-1110, R-1112and R-1113 are adjusted to a position commensurate with m;,, the m servowill continuously maintain its output shaft in a position commensuratewith m providing the required shaft output quantity to vary the power ofthe second anamorphoser via pinion 12-1.

The m servo of FIG. ll solves Expression (12). A potential proportionalto -lz derived as shown in FIG. 10 is applied to the input circuit ofthe m servo via terminal 1008 and summing resistor R-l123. A potentialproportional to I1 on terminal 1004 is multiplied by P by means ofpotentiometers R-l132 and R-1133, further multiplied by m by means ofpotentiometer R-1125, multiplied by m by means of potentiometer R-1116and applied to the input circuit of the m servo via summing resistorR-ll24. Since the quantities applied to the m servo will balance to zeroonly when potentiometer R-112S is adjusted commensurately with m the mservo continuously positions its output shaft in accordance with m;providing a shaft output to adjust the power m of the first anamorphoserto the required value via pinion 111.

The servo of FIG. 11 solves Expression (11). P0- tentials proportionalto m and l/m are derived with appropriate polarity of potentiometersR-117 and R-118 and summed in amplifier U-1111. The output potential ofamplifier U-llll is multiplied by sine 20 by means of resolver R-1103 toprovide a potential proportional to the left hand side of Equation (11),which potential is applied to the p servo via summing resistor R-1134.The shaft of resolver R-1103 is positioned in accordance with the angle20 by means of control knob 1101, a 1:2 gear reduction 1103 beingprovided as shown. The potentials proportional to the right hand side ofExpression (11) are derived by applying appropriate functions of theindependant variables via terminals variables via terminals 1011 and1013 and modifying such potentials by sin 25 and cos 2/3 by meansof-resolvers R-1137 and R-1138, respectively, the arms of which arepositioned by the output shaft of the p servo through a 1:2 gearreduction 1104. The 5 servo provides an output shaft quantity toposition the first anamorphoser at an angle ,9 with respect to theoriginal image via pinion 105.

Referring now to FIG. 12, there is shown in electrical schematic form anexemplary controller which may be used to operate a Type II embodimentof the invention such as that shown in FIG. 3, wherein the powers ofboth anamorphosers are varied and the angular orientation 0 of thesecond anamorphoser with respect to the first is varied, but where theangular orientation B of the first anamorphoser with respect to theoriginal image is maintained constant. A P control knob 1203 may bemanually set in accordance with the system spherical magnification,thereby adjusting the positions of potentiometers R-1201, R-1202,R-1203, R-1204, R-1205 and R-1206. A 5 control knob 1204 is provided toact through gear reducer 1201 to set the shafts of resolvers R-1231,R-1234, R-1236 and R-1237 in accordance with the angle 25. If desired,the gear reducer 1201 may be eliminated, and the dial or scale (notshown) used to set knob 1204 may he graduated in terms of 2s. The mservo of FIG. 12 receives input potentials which are functions of theindependent variables (k h a) and the constant settings of control knobs1203 and 1204 to solve the following expression, which may be obtainedby solving simultaneously particular ones of Equations (1) through (4),(5a), (6a) and (7) through (9).

cos 2B[i sin 26] A potential proportional to the first term on the righthand side of Equation (16) is obtained by applying the potential onterminal 1006 directly to the input circuit of the m servo via summingresistor R-1211 and by applying the same potential as modified by m bypotentiometers R-1215 and R-1216 to the input circuit via resistorR-1217. The second term on the right hand side of Expression (16) isapplied by modifying the 2 /h potential on terminal 1009 by P by meansof potentiometers R-1201 and R-1202, and by further modifying thepotential in accordance with m by means of potentiometers R-1218 andR-1219, to provide a servo input potential via summing resistor R-1220.The third term on the right hand side of the expression is derived bydividing the potential of terminal 1015 by P by means of potentiometersR-1203 and R-1204 and amplifier U-l, and this potential is applied viasumming resistor R-1210. The bracketed quantity on the left-hand side ofExpression (16) is derived by modifying the input potential fromterminal 1013 by sin 2,6 by resolver R-1234, by modifying the inputpotential from terminal 1012 by cos 2,) by means of resolver R-1231, andby combining the two potentials in summing amplifier U-1203. The outputquantity from amplifier U-1203 representing the bracketed quantity ismultiplied by (l-m in being applied to the m servo, the amplifier outputbeing connected directly to the m servo via summing resistor R-1209 andalso being connected indirectly via summing resistor R-1214 after havingbeen multiplied by m by potentiometers R-1212 and R-1213 and inverted inpolarity by amplifier U-1202. Since the input potentials applied to them servo input circuit cancel out to zero when and only when the outputshaft of the m servo has adjusted the arms of its potentiometers to aposition commensurate with m a shaft output is pro vided from the mservo to drive pinion 111 of FIG. 3 to vary the power of the firstanamorphoser as required. The m servo of FIG. 12 solves Equation (12).An I1 potential from terminal 1004 is multiplied by P by means ofpotentiometers R-1205 and R-1206, further

1. APPARATUS FOR ALTERING THE APPARENT PERSPECTIVE OF AN IMAGE OF ANOBJECT COMPRISING IN COMBINATION A PAIR OF ANAMORPHOSERS COAXIALLYDISPOSED ALONG AN OPTICAL AXIS, SAID PAIR OF ANAMORPHOSERS HAVING FOURCONTROL VARIABLES: M1, THE POWER OF A FIRST OF SAID ANAMORPHOSERS, M2,THE POWER OF A SECOND OF SAID ANAMORPHOSERS, B, THE ANGLE BETWEEN THEDIRECTION OF ANAMORPHIC POWER OF SAID FIRST OF SAID ANAMORPHOSERS AND AREFERENCE AXIS, AND 0, THE ANGLE BETWEEN THE DIRECTION OF ANAMORPHICPOWER